Section VO Vector Operations

In this section we define some new operations involving vectors, and collect some
basic properties of these operations. Begin by recalling our definition of a column
vector as an ordered list of complex numbers, written vertically (Definition CV).
The collection of all possible vectors of a fixed size is a commonly used set, so we
start with its definition.

When a set similar to this is defined using only column vectors
where all the entries are from the real numbers, it is written as
{ℝ}^{m} and is known as
Euclidean m-space.

The term “vector” is used in a variety of different ways. We have defined it as an
ordered list written vertically. It could simply be an ordered list of numbers, and written as
\left (2,\kern 1.95872pt 3,\kern 1.95872pt − 1,\kern 1.95872pt 6\right ). Or it could be
interpreted as a point in m
dimensions, such as \left (3,\kern 1.95872pt 4,\kern 1.95872pt − 2\right )
representing a point in three dimensions relative to
x,
y and
z axes.
With an interpretation as a point, we can construct an arrow from the origin to
the point which is consistent with the notion that a vector has direction and
magnitude.

All of these ideas can be shown to be related and equivalent, so keep that in
mind as you connect the ideas of this course with ideas from other disciplines. For
now, we’ll stick with the idea that a vector is a just a list of numbers, in some
particular order.

Subsection VEASM: Vector Equality, Addition, Scalar Multiplication

We start our study of this set by first defining what it means for two vectors
to be the same.

Now this may seem like a silly (or even stupid) thing to say so carefully. Of
course two vectors are equal if they are equal for each corresponding entry! Well,
this is not as silly as it appears. We will see a few occasions later where the
obvious definition is not the right one. And besides, in doing mathematics we need
to be very careful about making all the necessary definitions and making them
unambiguous. And we’ve done that here.

Notice now that the symbol ‘=’ is now doing triple-duty. We know from
our earlier education what it means for two numbers (real or complex)
to be equal, and we take this for granted. In Definition SE we defined
what it meant for two sets to be equal. Now we have defined what it
means for two vectors to be equal, and that definition builds on our
definition for when two numbers are equal when we use the condition
{u}_{i} = {v}_{i} for
all 1 ≤ i ≤ m.
So think carefully about your objects when you see an equal sign and think about
just which notion of equality you have encountered. This will be especially
important when you are asked to construct proofs whose conclusion states that
two objects are equal.

OK, let’s do an example of vector equality that begins to hint at the utility of
this definition.

Note the use of three equals signs — each indicates an equality of numbers (the
linear expressions are numbers when we evaluate them with fixed values of the
variable quantities). Now write the vector equality,

By Definition CVE, this single equality (of two column vectors) translates into
three simultaneous equalities of numbers that form the system of equations. So
with this new notion of vector equality we can become less reliant on referring to
systems of simultaneous equations. There’s more to vector equality than just this,
but this is a good example for starters and we will develop it further.
⊠

We will now define two operations on the set
{ℂ}^{m}. By
this we mean well-defined procedures that somehow convert vectors into
other vectors. Here are two of the most basic definitions of the entire
course.

So vector addition takes two vectors of the same size and combines them (in a
natural way!) to create a new vector of the same size. Notice that this definition is
required, even if we agree that this is the obvious, right, natural or correct way to
do it. Notice too that the symbol ‘+’ is being recycled. We all know how to add
numbers, but now we have the same symbol extended to double-duty and we use
it to indicate how to add two new objects, vectors. And this definition of our new
meaning is built on our previous meaning of addition via the expressions
{u}_{i} + {v}_{i}. Think about
your objects, especially when doing proofs. Vector addition is easy, here’s an example
from {ℂ}^{4}.

Our second operation takes two objects of different types, specifically a
number and a vector, and combines them to create another vector. In this
context we call a number a scalar in order to emphasize that it is not a
vector.

Notice that we are doing a kind of multiplication here, but we are defining a
new type, perhaps in what appears to be a natural way. We use juxtaposition
(smashing two symbols together side-by-side) to denote this operation rather than
using a symbol like we did with vector addition. So this can be another source of
confusion. When two symbols are next to each other, are we doing regular old
multiplication, the kind we’ve done for years, or are we doing scalar vector
multiplication, the operation we just defined? Think about your objects — if the
first object is a scalar, and the second is a vector, then it must be that we are
doing our new operation, and the result of this operation will be another
vector.

Notice how consistency in notation can be an aid here. If we write
scalars as lower case Greek letters from the start of the alphabet (such as
α,
β,
…) and write vectors in bold Latin letters from the end of the alphabet
(u,
v, …),
then we have some hints about what type of objects we are working with.
This can be a blessing and a curse, since when we go read another book
about linear algebra, or read an application in another discipline (physics,
economics, …) the types of notation employed may be very different and hence
unfamiliar.

Subsection VSP: Vector Space Properties

With definitions of vector addition and scalar multiplication we can
state, and prove, several properties of each operation, and some properties
that involve their interplay. We now collect ten of them here for later
reference.

Proof While some of these properties seem very obvious, they all require proof.
However, the proofs are not very interesting, and border on tedious. We’ll prove
one version of distributivity very carefully, and you can test your proof-building
skills on some of the others. We need to establish an equality, so we will do so by
beginning with one side of the equality, apply various definitions and theorems
(listed to the right of each step) to massage the expression from the left into the
expression on the right. Here we go with a proof of Property DSAC. For
1 ≤ i ≤ m,

Since the individual components of the vectors
(α + β)u and
αu + βu are equal
for all i,
1 ≤ i ≤ m, Definition CVE tells
us the vectors are equal. ■

Many of the conclusions of our theorems can be characterized as “identities,”
especially when we are establishing basic properties of operations such as those in
this section. Most of the properties listed in Theorem VSPCV are examples. So
some advice about the style we use for proving identities is appropriate right
now. Have a look at Technique PI.

Be careful with the notion of the vector
−u. This is a vector that we
add to u so that the result
is the particular vector 0.
This is basically a property of vector addition. It happens that we can compute
−u using
the other operation, scalar multiplication. We can prove this directly by writing
that